2100 CHAPTER 62. STOCHASTIC PROCESSES

≤ p∫

∞

0λ

p−1 1λ

∫X[X∗(T )>λ ]X (T )dPdλ

= p∫

Ω

X (T )∫ X∗(T )

0λ

p−2dλdP

= p∫

Ω

X (T )X∗ (T )p−1

p−1dP

≤ pp−1

(∫Ω

X∗ (T )p dP)1/p′(∫

Ω

X (T )p dP)1/p

Now divide both sides by (∫

ΩX∗ (T )p dP)1/p′

. Substituting Xτn for X(∫Ω

|Xτn∗ (t)|p dP)1/p

≤(∫

Ω

Xτn∗ (T )p dP)1/p

≤ pp−1

(∫Ω

Xτn (T )p dP)1/p

Now let n→ ∞ and use the monotone convergence theorem to obtain the inequality ofthe theorem. This establishes 62.9.32. The use of Fubini’s theorem follows from Lemma62.9.1.

Here is another sort of maximal inequality in which X (t) is not assumed nonnegative.

Theorem 62.9.5 Let {X (t)} be a right continuous submartingale adapted to the normalfiltration Ft for t ∈ [0,T ] and X∗ (t) defined as in Theorem 62.9.4

X∗ (t)≡ sup{X (s) : 0 < s < t} , X∗ (0)≡ 0,

P([X∗ (T )> λ ])≤ 1λ

E (|X (T )|) (62.9.33)

For t > 0, letX∗ (t) = inf{X (s) : s < t} .

ThenP([X∗ (T )<−λ ])≤ 1

λE (|X (T )|+ |X (0)|) (62.9.34)

AlsoP([sup{|X (s)| : s < T}> λ ])

≤ 2λ

E (|X (T )|+ |X (0)|) (62.9.35)

Proof: The function f (r)= r+≡ 12 (|r|+ r) is convex and increasing. Therefore, X+ (t)

is also a submartingale but this one is nonnegative. Also

[X∗ (T )> λ ] =[(

X+)∗(T )> λ

]and so from Theorem 62.9.4,

P([X∗ (T )> λ ]) = P([(

X+)∗(T )> λ

])≤ 1

λE(X+ (T )

)≤ 1

λE (|X (T )|) .